The quadratic formula is pretty easy to remember, and will
always work, even if the solutions to the quadratic equation aren't always real
numbers (they can be imaginary - the square roots of negative numbers). So
here it is:

_______
x = -b ± √b2 - 4ac
2a

First, just try it. Then we will do some more work with
it. Take the equation 2x2 + 7x + 6 = 0.

__________x = -7±√72
- 4(2)(6)
2(2)

x = -7 ± √1 4

x = -8/4 = -2ORx = -6/4 = -1.5

Try x2 - 2x + 4 = 0.

___________
x = -(-2)±√(-2)2 - 4(1)(4)
2(1)

___x = 4±√-12 2

As you can see, this equation has
no real solutions, because we need to take the square root of a negative number.

Now that you have seen these two
examples, it is time to see how the quadratic formula works, and it works on the
basis of the "completing the square" process.

So we have the standard form of the
equation: ax2 + bx + c = 0

Divide each term by a, to get x2
alone (you need this for completing the square):x2 + b x + c
= 0
a a

Then move the constant, c/a, to the other side by subtracting it
from both sides:x2 + b x = - c
a a

Now on the left side we want to complete the square. Remember
that that form is a2 + 2ab + b2. Basically, in
completing the square, to get the term of b2, when the coefficient of
x2 is 1, we just add the square of half the coefficient of x.
Ok...in a simpler showing: 1 * b =
b
2 a 2a

( b 2 =
b2
2a) 4a2

Add this ^^ to both sides.

x2 + b x +
b2 = - c + b2
a 4a2
a 4a2

We factor the left side and rearrange the right side.

(x + b )2 =
b2 - c 2a
4a2 a

Now get a common denominator for the right side. It will
be 4a2.

(x + b )2 =
b2 - 4ac 2a
4a2 4a2

(x + b )2 =
b2 - 4ac 2a
4a2

Take the square root of each side, but remember on the right
side to put plus OR minus.
_________x + b = ±√
b2 - 4ac 2a
4a2

(I can't show it well, but on the right side, it's the square
root of the WHOLE SIDE, not just the top line. Now simplify the right side
again.
________x + b = ±√
b2 - 4ac 2a
√4a2
________x + b = ±√
b2 - 4ac 2a
2a

Recognize this??? Well you should...*gives people who
don't glares*...just kidding. It really takes awhile to get the hang of
all this. You may have to go over this explanation more than a few times
to get it. You should know that it is easiest to factor a quadratic, if
possible, but this can also be used. Happy quadratic equation solving!!